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 Post subject: Rubiks cube positions?Posted: Tue May 21, 2013 9:24 am

Joined: Mon Feb 11, 2013 8:33 am
I've a question about the number of positions for a Rubik's cube. Is there a formula for the number of positions on a Rubik's cube. If there is a formula like that than my question is if its possible to calculate it for higher order cubes ( 4x4, 5x5 etc.)

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 Post subject: Re: Rubiks cube positions?Posted: Tue May 21, 2013 9:46 am

Joined: Sun Feb 17, 2008 2:32 pm
It's all about combinations

There are 8 corners that can be in 8, 7, 6, ..., 1 positions, so 8!
There are 12 edges that can be en 12, 11, 10, ..., 1 positions, so 12!
Corners can be in 3 different orientations, so 3^8
Edges can be in 2 different orientations, so 2^12

But not really... there are permutation restrictions, so you have to divide by 2. And there are orientation restrictions, so you have to divide by 2*3.

result: 8! * 12! * 3^7 * 2^11 / 2 = 43.252.003.274.489.856.000 different positions

http://en.wikipedia.org/wiki/Rubik's_Cube#Permutations

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 Post subject: Re: Rubiks cube positions?Posted: Tue May 21, 2013 10:37 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
We had a thread about counting the permutations of the Dioctipoid here: viewtopic.php?f=1&t=22891 Some of the content there could be useful.

There was a discussion about the N-Pyraminx and N-FTO permutation calculations here:
viewtopic.php?p=287293#p287293

And I posted a formula for the N-minx here:
viewtopic.php?p=285637#p285637

Have you tried to calculate the positions for an NxNxN cube yet? If not I can walk you through it.

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 Post subject: Re: Rubiks cube positions?Posted: Wed May 22, 2013 6:41 am

Joined: Mon Feb 11, 2013 8:33 am
bmenrigh wrote:
Have you tried to calculate the positions for an NxNxN cube yet? If not I can walk you through it.

First thank you!! But I haven't calculated the positions for an nxnxn cube?
Max Knoop

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3x3: 21.09
2x2: 3.00

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 Post subject: Re: Rubiks cube positions?Posted: Wed May 22, 2013 7:03 am

Joined: Sun Feb 20, 2011 5:14 pm
Location: Hertfordshire, UK
cisco wrote:
result: 8! * 12! * 3^7 * 2^11 / 2 = 43.252.003.274.489.856.000 different positions

Out of curiosity, does this number exclude identical but 'rotated' positions? In other words the same position of cubes in relation to eachother, but with the cube upside down for example.

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 Post subject: Re: Rubiks cube positions?Posted: Wed May 22, 2013 9:21 am

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
cisco wrote:
result: 8! * 12! * 3^7 * 2^11 / 2 = 43.252.003.274.489.856.000 different positions

Out of curiosity, does this number exclude identical but 'rotated' positions? In other words the same position of cubes in relation to eachother, but with the cube upside down for example.
If you fix the 6 centres in space, you can permute edges and corners and re-orientate them relative to this position in space in the ~43*10^18 different configurations.
The 24 different orientations in space are not counted as different positions.

BTW, on a 3x3x3 Supercube (showing the orientations of all 6 centres) you have to multiply the calculated figure 43252003274489856000 by 2048 (4^6/2 ; divided by 2 because the last centre can be re-orientated by 180° degrees only, independent on the other 5. (btw, never trust Excel doing large integer calculations. In this case you will get 43.252.003.274.489.900.000. Gus pointed me to http://www.wolframalpha.com for correct calculations )

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 Post subject: Re: Rubiks cube positions?Posted: Wed May 22, 2013 11:44 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
cisco wrote:
result: 8! * 12! * 3^7 * 2^11 / 2 = 43.252.003.274.489.856.000 different positions

Out of curiosity, does this number exclude identical but 'rotated' positions? In other words the same position of cubes in relation to eachother, but with the cube upside down for example.

To echo what Konrad said, the answer is "yes" under some assumptions.

For any of the 4.3 * 10^19 positions. If you pick up that cube, flip it around at random, and set it down, the resulting position is identical and only counted once.

The reason is that the ((8! * 12!) / 2) * (2^12 / 2) * (3^8 / 3) calculation doesn't take into account centers. The center core can be in 24 orientations but because the core shows the orientation of the puzzle we can ignore it.

If you want to treat the centers as identical (like the Void Cube) you need to divide the 4.3 * 10^19 value by 12. The reason this is 12 and not 24 has to do with the parity of the edges and corners.

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Rubiks cube positions?Posted: Wed May 22, 2013 11:50 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Thedutchcuber wrote:
bmenrigh wrote:
Have you tried to calculate the positions for an NxNxN cube yet? If not I can walk you through it.

First thank you!! But I haven't calculated the positions for an nxnxn cube?
Max Knoop

Have you tried? Are you able to calculate the positions for the 3x3x3? If so, can you calculate the number of positions for the 4x4x4 and 5x5x5?

If you can detail what you know and can do it'll give us a better idea of where to start.

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Rubiks cube positions?Posted: Wed May 22, 2013 3:30 pm

Joined: Sun Apr 16, 2000 3:17 pm
Location: Hong Kong
After an adult beverage... I read the title of this and thought....

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 Post subject: Re: Rubiks cube positions?Posted: Wed May 22, 2013 3:51 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
katsmom wrote:
After an adult beverage... I read the title of this and thought....

katsmom bringing new meaning to terms like "superflip" and "finger tricks" since 2000

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